Predicting the thermal conductivity of inorganic and polymeric glasses:

The role of anharmonicity

Sergei Shenogin, Arun Bodapati, and Pawel Keblinski

Rensselaer Nanotechnology Center and Materials Science and Engineering Department,

Rensselaer Polytechnic Institute, Troy, NY 12180, USA

Alan J. H. McGaughey

Department of Mechanical Engineering,

Carnegie Mellon University, Pittsburgh, PA, 15213, USA

Abstract

The thermal conductivity of several amorphous solids is numerically evaluated within the

harmonic approximation from Kubo linear-response theory following the formalism developed

by Allen and Feldman.12 The predictions are compared to the results of molecular dynamics

(MD) simulations with realistic anharmonic potentials and to experimental measurements. The

harmonic theory accurately predicts the thermal conductivity of amorphous-silicon, a model

Lennard-Jones glass, and a bead-spring Lennard-Jones glass. For polystyrene and amorphous-

silica at room temperature, however, the harmonic theory underestimates the thermal

conductivity by a factor of about two. This result can be explained by the existence of additional

thermal transport via anharmonic energy transfer. More surprisingly, the thermal conductivity of

polystyrene and amorphous-silica at low temperature (MD and experimental) is significantly

below the predictions of the harmonic theory. Potential reasons for the failure of the harmonic

theory of disordered solids to predict the thermal conductivity of glassy polymers are discussed.

I. Introduction

The temperature, T, dependence of the thermal conductivity, k, of a broad range of inorganic and

polymeric glasses is very similar, showing a monotonic increase with increasing temperature.1

Despite a highly disordered structure, at temperatures below 10 K the thermal energy is carried

by polarized and delocalized vibrations, i.e., propagating phonons. Consequently, below 10 K, k

~ T2 is observed, a trend usually attributed to the phonon scattering in a two-level system.2 In the

temperature range of 10-20 K, a plateau is observed, followed by further increase in the thermal

conductivity at higher temperature, which has been attributed to excitations of higher frequency

vibrations and/or anharmonic effects.3 Contrary to the behavior of dielectric crystals, there is no

decrease in the thermal conductivity at higher temperatures caused by a reduction of phonon

mean free path by phonon-phonon scattering.

The thermal conductivity of a glass can be interpreted in terms of the Boltzmann

transport theory for a phonon gas in a dielectric:4

∑=i

iii lvCV

k31 , (1),

where the summation is performed over all vibrational modes, Ci is the spectral heat capacity, vi

is the group velocity, and li is the phonon mean free path. A temperature-dependent average

phonon mean free path may be introduced to fit experimental data in the plateau region,

following the idea that phonons with energy below the mobility edge can propagate through the

structure.5 In glasses, with the exception of very long-wavelength modes, the vibrational modes

are not polarized and do not propagate like phonons in a crystal. Therefore, Eq. (1) can only

describe the few low-frequency acoustic modes for which a non-zero group velocity can be

assigned. However, when Eq. (1) is used to estimate thermal conductivity in a glass, one

typically assumes that the group velocity for all the modes is equal to the speed of sound and the

mean free path is given by the average interatomic distance. This approach can give qualitatively

correct results above the plateau region, with the temperature dependence of the thermal

conductivity entirely explained by the increase of the heat capacity at higher temperatures.6

However, because lattice dynamics calculations clearly show that the majority of the vibrational

modes in a glass are nonpropagating,7 Eq. (1) will be used here only for qualitative comparison

between glasses and crystals.

The thermal conductivity of a crystal decreases at higher temperature due to the reduction

of phonon mean free path, which is caused by increased phonon-phonon scattering. The

underlying mechanism is the strain-dependence of the elastic constants of the crystal, which

results from the anharmonicity of the atomic interactions.1 Unlike in a crystal, the thermal

conductivity of a glass increases monotonically with increasing temperature, suggesting that the

role of anharmonicity is different. Heat transfer in a glass can be analyzed in terms of energy

hopping between localized non-propagating vibrational modes,3,8,9,10,11 with anharmonicity

contributing positively to the thermal conductivity, and its contribution increasing with

increasing temperature. Therefore, the increase of thermal conductivity with increasing

temperature above the plateau region results not only from the increased heat capacity of the

vibrational modes, but also from stronger anharmonic coupling between the modes.10

Allen and Feldman put forward an alternative mechanism based on the “diffusion” of

energy between the extended non-propagating vibrational modes (AF theory).7,12 They applied

the energy flux operator method13 to an amorphous structure and calculated the thermal

conductivity within the harmonic approximation. The AF theory successfully predicts the

temperature dependence of thermal conductivity in amorphous silicon14,15 and Lennard-Jones

(LJ) glasses,16 but the applicability of AF theory for other amorphous materials such as polymers

has not been studied. Because the majority of vibrational modes in amorphous-Si and LJ glasses

are spatially extended and non-polarized, it is perhaps not surprising that the harmonic theory

works well. For amorphous solids with a large fraction of localized modes, energy hopping may

be applicable.

To investigate the roles of harmonic and anharmonic transport channels, and the related

localization of vibrations on the thermal conductivity of glasses, we here use molecular dynamics

(MD) simulation and several model amorphous systems including amorphous Si, amorphous

silica, and an amorphous polymer. We note that in MD simulations realistic interaction potentials

that include anharmonic effects are used, while the AF theory considers only harmonic energy

transfer. The results of the MD simulations are compared with the predictions of the AF theory.

II. Simulation Setup and Thermal Conductivity Prediction

Five model amorphous structures were generated for our study: (i) 600 identical beads

interacting by a 12-6 LJ potential, (ii) the 600 beads combined into 15 linear chains with 40

beads in each, and the FENE interaction potential within the chain,17 (iii) amorphous silicon,

with 512 atoms interacting by the Stillinger-Weber potential18, (iv) amorphous silica, with 288,

576, or 972 atoms interacting by the BKS19 potential, and (v) an atactic polystyrene model

consisting of 1, 2, or 3 chains (338, 676, or 1014 explicit carbon and hydrogen atoms), prepared

according to conformational statistics at room temperature20 and further equilibrated at standard

temperature and pressure using the Polymer Consistent Force Field (PCFF).21 The choice of the

structures includes the most popular simplified glass models [(i) and (ii)], and realistic models of

inorganic [(iii) and (iv)] and organic (v) glasses with well established forcefields for each

structure. All simulations have periodic boundary conditions applied in all three dimensions.

Snapshot images of the five studied models are shown in Fig.1.

The theoretical calculation of the thermal conductivity is done according to the AF

theory:7

ii

i DTCV

k )(1 ∑= (2)

where V is the volume, and the sum is over all vibrational modes. The thermal diffusivity of each

mode, Di, is calculated with the heat current operator Sij with summation over all other

vibrational modes12

)(3

2

2

2

jiij

iji

i SVD ωωδω

π−= ∑

≠

, (3)

where ħ is the Planck constant divided by 2π and ωi is the angular frequency of mode i. For our

finite-size systems with periodic boundaries, the δ-function in Eq. (3) is broadened to a

Lorentzian-shaped peak with a width exceeding the mean energy-level spacing.12 The elements

Sij are calculated from the mass-scaled force constant matrix, as shown in Appendix A of the

Allen and Feldman paper.12 For all structures, a careful energy minimization was done prior to

the calculation of the force constant matrix and the vibrational mode analysis. Because the force

constant matrix depends only on a quadratic expansion of the potential energy around its

minimum, we will refer to the results of these calculations as the thermal conductivity under the

harmonic approximation. Within this approximation, the temperature dependence of the thermal

conductivity will only be a result of the temperature dependence of the heat capacity. If the

predictions of the harmonic approximation are to be compared with the results of classical MD

simulations, one should use the classical limit for the heat capacity, Ci=kB, where kB is the

Boltzmann constant, in Eq. (2).

Since the non-zero heat current operator in Eq. (3) requires that two modes overlap in

space, the thermal diffusivity of a mode strongly depends on its degree of localization. In fact,

the diffusivity of a localized mode calculated by Eq. (3) is strictly equal to zero. In order to

characterize the degree of mode localization we calculated the inverse of the participation ratio

pi:

[ ]∑=atomsAll

aai

ip221 ε (5)

where εai are the components of the normalized polarization vectors for each mode,

which satisfy:

∑ =atomsAll

aai 12ε . (6)

By comparing pi and Di for different modes, we conclude that modes with participation ratio

greater than 0.3 - 0.4 are not localized.

To predict the thermal conductivity from non-equilibrium MD (NEMD), the length of the

simulation cell in one direction for each structure was multiplied by six. The structure was then

equilibrated at a specified temperature, and a constant heating power W was applied to the center

of the structure, and the same constant cooling power -W was applied to the periphery. Once a

constant temperature gradient T∇ is reached, the thermal conductivity is calculated according to

the Fourier law:22

AT

Wk∇

= , (7)

where A is the effective heat flux area (double the area of the cell face normal to the flux

direction). Additional simulations were performed on longer simulation cells to make sure that

no size effects exist at these lengths for NEMD simulations. For the amorphous silica structure,

the thermal conductivity is predicted from equilibrium MD and the Green-Kubo method.23

III. Results:

A comparison of the results for the LJ-glass and the LJ-glass with particles grouped in linear

chains is presented in Fig. 2. The normalized vibrational density of states, plotted vs. reduced

frequency in Fig. 2, top plot, shows that polymerization shifts approximately 1/3 of the

vibrational modes to higher frequency. This result simply reflects the fact that for the linear

chain, two of the three degrees of freedom associated with each bead are unaffected by bonding,

while the frequencies of the modes associated with intra-chain motion increase. As seen in Fig.

2, middle plot, the participation ratio of the high frequency modes is less then 0.15, indicating

that that these modes are localized in contrast to the rest of the spectrum, which has participation

ratio ~0.5. The analysis of the associated eigenvectors shows that each of the high-frequency

modes is associated with the oscillation of several chain segments along the local chain direction

(i.e., longitudinal modes). The thermal diffusivity of the high frequency modes, evaluated from

AF theory, is negligible compared to the diffusivity of lower-frequency modes (Fig. 2, bottom

plot). As a result, the thermal conductivity calculated from AF theory for the bead-spring model

is almost exactly 2/3 of that of the original LJ glass (0.102 W/m-K compared to 0.150 W/m/K).

These predictions agree with the results of the NEMD simulations (0.097 W/m-K and 0.142

W/m/K). The major conclusion from these results is that the harmonic approximation for thermal

conductivity works very well for these model LJ systems. The other important observation is

that, contrary to intuition, the introduction of strong covalent bonds reduces the thermal

conductivity of the structure. This latter observation is perhaps associated with the fact that the

chains are fully flexible, and thus do not allow for the possibility of rapid energy transport along

the straight chain segments. This observation, however, is valid only for chemically uniform

glasses in which the fraction of localized vibrations is negligible and the mechanism of thermal

transport is primarily diffusive as suggested by AF theory. As we will show, for glasses with

complex chemical structure, like silicates and aluminosilicates, the diffusive mechanism does not

work and the experiments show the opposite tendency (i.e., glasses with stronger covalent

bonding exhibit higher thermal conductivity).24

The results of the vibrational mode analysis for amorphous silicon are plotted in Fig. 3.

The density of states for amorphous silicon consists of two characteristic peaks and compares

favorably with previously reported results.7,25 The spectrum consists mostly of delocalized

vibrational modes (participation ratio p~0.5), which will contribute to the thermal conductivity.

Only a small fraction of the highest frequency modes are localized and have essentially zero

diffusivity. A comparison of the thermal conductivity calculated from AF theory with the

predictions of the NEMD simulations is shown in Fig. 4. Two important observations can be

made: (i) the classical MD simulations show no temperature dependence for the thermal

conductivity, and (ii) the constant value of the thermal conductivity obtained by NEMD is very

close to the classical limit for harmonic theory (0.91 W/m/K and 0.85 W/m/K). As previously

reported by Allen and Feldman,7,12 the harmonic theory calculations are close to the experimental

results. The temperature independent value obtained by NEMD is close to the experimental

values at temperatures above 100 K. These results simply restate the Allen and Feldman

conclusion that the anharmonicity-related hopping mechanism between localized modes does not

play an important role in the thermal conductivity of amorphous silicon.

The results for amorphous silica are quite different (see Figs. 5 and 6). The high-

frequency part of the vibrational spectrum (ν>30 THz) consists mostly of localized modes

(p<0.2). These localized modes have thermal diffusivities close to zero. Contrary to amorphous

silicon, the thermal conductivity obtained from classical MD simulations is temperature-

dependent and at room temperature is about 30% larger than that predicted by AF theory (solid

line in Fig. 6). This result indicates that anharmonic transport mechanisms are more important in

amorphous silica than in amorphous silicon. These mechanisms are likely associated with

anharmonic energy transfer between localized vibrational modes, or between localized and

delocalized modes. The contribution from energy hopping increases with temperature, and after

the addition of a temperature-independent harmonic thermal conductivity, can qualitatively

explain the results of the classical MD simulations. In particular, the MD-measured thermal

conductivity of amorphous silica shows the linear temperature-dependence often found in

polymers and other glasses with complex composition.3,10

What is quite surprising, however, is that the predictions of the AF theory in the classical

limit [Ci=kB in Eq. (2)] show strong disagreement with the results of MD at low temperatures,

where the harmonic theory should describe the behavior of the system reasonably well. In

particular, the thermal conductivities obtained from MD fall below the AF predictions at

sufficiently low temperatures. This result is contrary to the expectation that the harmonic

contribution to the thermal conductivity should be temperature independent for a classical system

and converge to a constant value at low temperature. In addition, we observed that the AF

prediction is dependent on the sample size, with larger values predicted for larger samples. In

particular, the classical limit yields thermal conductivities of 0.94, 1.33 and 1.64 W/m-K for 288,

576, 972 atom samples. By contrast, the results of the AF theory for amorphous silicon are

essentially size independent.7,14 A similar size dependence was previously reported from NEMD

simulations on a silica glass, while in the same time no strong size dependence was found for

amorphous selenium.26 This finding is quite surprising, considering that the larger number of

localized states in the vibrational spectra of chemically non-uniform materials should be less

affected by the finite size of a system.

A similarly puzzling behavior is found for amorphous polystyrene. The results of the MD

simulations and vibrational mode analysis for the amorphous polystyrene model are presented in

Figs. 7 and 8. These results show similar trends as the results for amorphous silica, with an even

larger fraction of localized modes (because the ratios of the masses and force constants between

the different atom types are larger). In polystyrene, all modes with a frequency above 5 THz are

localized and contribute very little to the diffusive thermal conductivity. As a result, almost all of

the diffusive thermal conductivity comes from the very small low-frequency part of the

vibrational spectrum, which has non-zero thermal diffusivity (see Fig. 7, bottom plot). This

finding is in contrast to the behavior of the flexible LJ chain described above, where only 1/3 of

the modes were localized and the rest of the modes were delocalized, as in the LJ glass, or in

amorphous silicon. It appears that the main difference between the flexible chains and

polystyrene is the chain rigidity, which significantly increases the fraction of localized modes

and leads to mode localization on a larger number of atoms along rigid parts of the chain. As can

be seen from Fig. 8, as is the case for amorphous silica, at low temperature the classical limit of

the thermal conductivity from the AF theory is significantly higher then the results of the MD

simulations. This result again shows the puzzling behavior of the harmonic theory overpredicting

the thermal conductivity at low temperatures. Also, as for amorphous silica, a size dependence of

the AF predictions is found.

IV. Conclusions

From the comparison between the predictions of MD simulations and the AF theory of

disordered harmonic solids, several observations can be made. First, the harmonic approximation

works well for amorphous materials with one type of atom and identical interaction potentials

between all atoms (e.g., a simple LJ glass and amorphous silicon). For these systems,

anharmonicity does not play an important role because the number of localized modes is

negligible and the major mechanism of energy exchange between extended modes is harmonic in

nature and can be well described by AF theory. Second, for systems with forced “anisotropy,”

like the linear LJ bead-spring model, the harmonic approximation may still work as long as the

energy hopping mechanism is restricted, for example, because of the geometry of localized

vibrations. Finally, for systems with complex composition like amorphous silica or an organic

polymer glass, anharmonic coupling between localized modes, or between localized and

delocalized modes is the major mechanism of heat conduction. More interesting and puzzling is

the fact that the harmonic theory overpredicts the MD thermal conductivities at low temperatures

for amorphous silica and polystyrene. Therefore, our results indicated that with strong

anharmonicity, the harmonic theory of disordered solids is not applicable, even in the low-

temperature limit.

Figure Captions

Fig.1 Snapshot images of five amorphous molecular models: (a) LJ beads, (b) bead-spring

model, (c) amorphous silicon, (d) amorphous silica, and (e) atactic polystyrene.

Fig.2 Vibrational density of states, participation ratio, and mode diffusivity plotted as a function

of reduced frequency for LJ bead and bead-spring models.

Fig.3 Vibrational density of states, participation ratio, and mode diffusivity plotted as a function

of frequency for amorphous silicon.

Fig.4 Temperature dependence of thermal conductivity for amorphous silicon predicted by

Allen-Feldman theory (solid line), calculated from NEMD (solid circles) and obtained in

experiments (triangles27 and crosses28).

Fig.5 Vibrational density of states, participation ratio, and mode diffusivity plotted as a function

of frequency for amorphous silica.

Fig.6 Temperature dependence of thermal conductivity for amorphous silica predicted by Allen-

Feldman theory with different number of atoms in the model (solid lines), calculated from

NEMD (circles) and obtained in experiments (triangles29).

Fig.7 Vibrational density of states, participation ratio, and mode diffusivity plotted as a function

of frequency for atactic polystyrene.

Fig.8 Temperature dependence of thermal conductivity for atactic PS predicted by Allen-

Feldman theory with different number of atoms in the model (solid lines), calculated from

NEMD (circles) and obtained in experiment30 (triangles).

(a)

(b)

(c)

(d)

(e)

Fig.1

0.0

0.2

0.4

Beads + Springs

DO

S*

Beads

0.0

0.5

1.0

Beads + SpringsBeads

PR

0 2 4 6 8 100

4

8

Beads + Springs

Beads, Beads + Springs

ν∗

Di[(l

*)2 /t*

]

Fig.2

0.00

0.04

0.08D

OS

0.0

0.5

1.0

PR

0 5 10 15 20 250.00

0.01

0.02

Di[ c

m2 /s

]

ν [THz]

Fig.3

0 100 200 3000.0

0.5

1.0

1.5 Amorphous SiK

[W/m

/K]

T [K]

Fig.4

0.0

0.5

1.0

PR

0 10 20 30 400.00

0.01

0.02

Di[ c

m2 /s

]

ν [THz]

0.00

0.04

0.08

DO

S

Fig.5

0 100 200 3000.0

0.5

1.0

1.5

2.0

2.5

972 atoms

288 atoms

576 atoms

Amorphous SiO2

K [W

/m/K

]

T [K]

Fig.6

0.00

0.04

0.08D

OS

0.0

0.5

1.0

PR

0 20 40 60 80 1000.00

0.01

0.02

Di[ c

m2 /s

]

ν [THz]

Fig.7

10 100 10000.00

0.05

0.10

0.15

0.20

676 atoms

1014 atoms

338 atoms

Polystyrene

K [W

/m/K

]

T [K]

Fig.8

References 1 J.E.Parrot, A.D.Stuckes Thermal Conductivity of Solids, Pion Ltd, 1975 2 P.W.Anderson et al Phys.Rev., 25, 1, (1972) 3 A.Jagannathan et al. Phys.Rev.B, 39,13465 (1989) 4 R. E. Peierls “Quantum Theory of Solids” Oxford Univ.Press, London, 2004 5 R.C.Zeller and R.O.Pohl Phys.Rev.B, 4,2029 (1971) 6 C.Kittel, Phys.Rev. 75, 972 (1948) 7 J.L.Feldman et al. Phys.Rev.B, 48,12589 (1993) 8 V.G.Karpov and D.A.Parshin, Pis’ma Zh.Eskp.Teor.Phys. 38, 536 (1983); JETP Lett. 38,648

(1983) 9 A.Jagannathan et al. Phys.Rev.B, 41,3153 (1990) 10 S.Alexander et al., Phys.Rev.B 34, 2726 (1986) 11 D. G. Cahill, S. K. Watson, R. O. Pohl, Phys. Rev. B 46, 6131 (1992).

12 P.B.Allen and J.L.Feldman, Phys.Rev.B, 48,12581 (1993) 13 R.J.Hardy Phys.Rev., 132,168 (1963) 14 J.L.Feldman et al, Phys.Rev.B, 59,3551 (1999) 15P.B.Allen et al, Phil.Mag. B, 79, 1715 (1999) 16 P.B. Allen, SUNY Stony Brook, private communication. 17 K. Binder, A. Milchev and J. Baschnagel. Ann. Rev. Mater. Sci. 26 (1996) 18 F. H. Stillinger and T. A. Weber Phys. Rev. B, 31, 5262 (1985). 19 B.W.H. Van Beest, G.J. Kramer and R.A. Van Santen, Phys. Rev. Lett. 64, 1955 (1990). 20 M. Muller et al J. Chem. Phys. 114, 9764, (2001) . 21 J.R.Mapple et al J. Comput. Chem., 15, 162 (1994); Sun, H. Macromolecules 28, 701, (1995). 22 P.K. Schelling et al Phys.Rev.B, 65, 144306 (2002). 23 A. J. H. McGaughey and M. Kaviany, Int. J. Heat Mass Transfer 47 (2004) 1799 24 D.G.Cahill et al, Phys.Rev.B, 44,12226 (1991) 25 W.A.Kamitakahara et al, Phys.Rev.B, 36,6539 (1987) 26 C.Oligschleger and J.C.Schoen, Phys.Rev.B, 59, 4125 (1999) 27 G. Pompe and E. Hegenbarth, Phys.Status Solidi B 147, 103 (1988). 28 D.G.Cahill et al, J.Vac.Sci.Technol.A, 7, 1259 (1989)

29 D.H.Damon Phys.Rev.B, 8, 5860 (1973) 30 T.Kikuchi el al , J.Macromol.Sci B, 42, 1097 (2003)